Actually, you need to talk to the fisherman in Balfonheim twice (until he talks about his rod) then go all the way to the Aerodome (but don't exit the area the fisherman is in) then run back to where he was standing. The fishing rod will be leaning against the wall.

For more detail on the fishing game itself, do a forum search for fishing.

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He who fights with monsters might take care lest he thereby become a monster. For when you gaze for long into the abyss, the abyss gazes also into you.

your also going to need a new rod,the bottles that you find from fishing give you clues where to find it,you'll meet gilgamesh at the end of it but he doesn't fight you,he hands you the rod.You must get the better rod to fish in the better areas or you won't complete the fishing side-quest.

As you do the fishing game you'll gain coloured bottles... each coloured bottle needs to be taken to a certain location to make it shatter. Once they shatter you'll receive a coloured message in your inventory. Get all of the messages, then you'll be able to find:

SPOILER

Gilgamesh, at the end of the Zeviah Span (the area that's isolated from the rest of the Span. He'll give you the upgraded fishing pole.

For more detailed info on where to take the bottles, try searching the forum for the keyword fishing.

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He who fights with monsters might take care lest he thereby become a monster. For when you gaze for long into the abyss, the abyss gazes also into you.

red bottle :mosphoran highwaste, babling vale, look for a faint glow near the shrine in the northwest

yellow bottle :cerrobi steppe, turn on the windmills number 1,2,5,7,9

black bottle :garamsythe waterway, look for a faint glow around the suice gates

after that, you can go to barheim and meet gilgamesh.. then you can continue fishing and finish it

From another thread, but the most concise details about what to do with the coloured bottles.

The final note for the fishing game is that you have to beat the River Lord 9 times in order to catch him and get Lu Shang's Badge, which is the end goal of the entire freaking side quest. And it can get hard... personally, I get comfortable with the order (it always stays the same for the River Lord), and once I've got it down I close my eyes and just enter away.

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He who fights with monsters might take care lest he thereby become a monster. For when you gaze for long into the abyss, the abyss gazes also into you.